Prokaryotic Gene Structure
Prokaryotic genes have a simple one-dimensional structure
5’→
5’→
←5’
←5’
ATG AAA ATG GCA . . . GCA TTG CTA TAG
Start
codon
Stop
codon
Note that the ATG codon encodes both start and methionine
Prokaryotic Gene Structure
Prokaryotic gene prediction begins with ORF finding
ATG AAA GCA ATG . . . GCA TTG CTA TAG
Possible
start
Alternate
start
Stop
codon
Because of the possibility of alternate start
sites, it’s not unusual for several ORFs to share
a common stop codon
An ORF finder needs to be able to find overlapping ORFs, whether they
end with the same stop codon, or overlap in a different frame
Prokaryotic Gene Structure
Prokaryotic gene prediction begins with ORF finding
'ATG(...)*?(TAA|TAG|TGA)'
A regular expression crafted to find ORFs
must also exhibit “non-greedy” behaviour
Note that many bacteria also employ rarer alternate start codons, most
commonly GTG and TTG. But we’ll pretend this doesn’t happen!
Higher Order Markov Chains
We don’t need to always just consider the most recent state
An nth order Markov process is a
stochastic process where the probabilities
associated with an event depend on the
previous n events in the state path
𝑷 𝒙𝒊 𝒙𝒊−𝟏 , 𝒙𝒊−𝟐 , … , 𝒙𝟏 = 𝑷(𝒙𝒊 |𝒙𝒊−𝟏 , … , 𝒙𝒊−𝒏 )
So far all the Markov models we have seen so far have been of order 1
In the case of a first order process this statement
reduces to our statement of the Markov property
Higher Order Markov Chains
Higher order models have an equivalent first order model
An nth order Markov chain over alphabet A
is equivalent to a first order Markov chain
over the alphabet An of n-tuples…
𝑷(𝒙𝒊 |𝒙𝒊−𝟏 , … , 𝒙𝒊−𝒏 ) =
𝑷 𝒙𝒊 , 𝒙𝒊−𝟏 , … , 𝒙𝒊−𝒏+𝟏 𝒙𝒊−𝟏 , … , 𝒙𝒊−𝒏
Practically, this says we can implement a higher order model just by
expanding the alphabet size of a first order model
This follows trivially from P(X,Y|Y) = P(X|Y)
Higher Order Markov Chains
Consider this first order Markov process
A = {A, B}
A
e
S
B
Here the alphabet A (our set of states)
consists of just A and B
How would we convert this to a second order Markov process?
Higher Order Markov Chains
Now reconfigured as a second order model
AA
AB
BA
BB
A = {AA, AB,
BA, BB}
Note how we have disallowed certain transitions (i.e. set
their probability to zero). Start and End omitted for clarity
Inhomogeneous Markov Chains
A Markov model of genes should model codon statistics
ATG AAA GCA GTC
1
2
3
In true coding genes, each of the three positions
within a codon will be statistically distinct
This can be accomplished in a few different ways…
Inhomogeneous Markov Chains
A Markov model of genes should model codon statistics
ATG AAA GCA GTC
1
2
3
𝟏
𝟐
𝟑
𝟏
𝟐
𝟑
𝒂𝒙𝟏 𝒙𝟐 𝒂𝒙𝟐 𝒙𝟑 𝒂𝒙𝟑 𝒙𝟒 𝒂𝒙𝟒 𝒙𝟓 𝒂𝒙𝟓 𝒙𝟔 𝒂𝒙𝟔 𝒙𝟕
...
One idea is to intersperse three different Markov chains in alternating fashion
This can also be recast as an HMM with additional states in obvious way
Histograms with matplot lib
We’ll use this to look at log-odds per NT distributions
import numpy as np
import pylab as P
. . .
# probs here should be your list of probabilities
# 50 here corresponds to the number of desired
n, bins, patches = P.hist(probs, 50, normed=1, histtype='stepfilled')
P.setp(patches, 'facecolor', 'g', 'alpha', 0.75)
P.show()
More histogram examples may be found at:
matplotlib.org/examples/pylab_examples/histogram_demo_extended.html